The idea that the world is ending on December 21, 2012, comes from a misinterpretation of the Mayan calendar. Because their calendar ends on the 21st, some believe the world must be ending, as well.

This idea has become so widespread in popular culture that NASA Science has made repeated efforts to debunk it. They've now even gone so far as to produce a video for the 22nd of December, explaining (again) why the world didn't end:

Jim Towey -Served as an Assistant to the President of the United States & the Second president of Ave Maria University
Jeff Fortenberry (R-NE) - Nebraska's 1st District
Michelle Joni - Mastermind of the Preschool for Adults NY